Résumé: The asymptotic robustness of estimators as a function of a rarity parameter, in the context of rare-event simulation, is often qualified by properties such as bounded relative error (BRE) and logarithmic efficiency (LE), also called asymptotic optimality. However, these properties do not suffice to ensure that moments of order higher than one are well estimated. For example, they do not guarantee that the variance of the empirical variance remains under control as a function of the rarity parameter. We study generalizations of the BRE and LE properties that take care of this limitation. They are named bounded relative moment of order k (BRM-k) and logarithmic efficiency of order k (LE-k), where k . 1 is an arbitrary real number.We also introduce and examine a stronger notion called vanishing relative centered moment of order k, and exhibit examples where it holds. These properties are of interest for various estimators, including the empirical mean and the empirical variance. We develop (sufficient) Lyapunov-type conditions for these properties in This work has been supported by NSERC-Canada grant No. ODGP0110050 and a Canada Research Chair to P. L’Ecuyer,NSFgrantDMS0595595 to J. H. Blanchet, EuroFGI Network of Excellence and INRIA’s cooperative research initiative RARE to B. Tuffin, and the Binational Science Foundation Grant 2002284 to P. W. Glynn. P. L’Ecuyer, thanks the IRISA for its generous support during his sabbatical in Rennes, where much of this article was written

Langue: Anglais